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mathematical optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
, the method of Lagrange multipliers is a strategy for finding the local
maxima and minima In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
of a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
subject to equality constraints (i.e., subject to the condition that one or more
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
s have to be satisfied exactly by the chosen values of the variables). It is named after the mathematician
Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiaderivative test In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about ...
of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function. The method can be summarized as follows: in order to find the maximum or minimum of a function f(x) subjected to the equality constraint g(x) = 0, form the Lagrangian function :\mathcal(x, \lambda) = f(x) + \lambda g(x) and find the
stationary point In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" inc ...
s of \mathcal considered as a function of x and the Lagrange multiplier \lambda; this means that all
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
s should be zero, including the partial derivative with respect to \lambda. The solution corresponding to the original constrained optimization is always a
saddle point In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function ...
of the Lagrangian function, which can be identified among the stationary points from the
definiteness In linguistics, definiteness is a semantic feature of noun phrases, distinguishing between referents or senses that are identifiable in a given context (definite noun phrases) and those which are not (indefinite noun phrases). The prototypical d ...
of the bordered Hessian matrix. The great advantage of this method is that it allows the optimization to be solved without explicit
parameterization In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, de ...
in terms of the constraints. As a result, the method of Lagrange multipliers is widely used to solve challenging constrained optimization problems. Further, the method of Lagrange multipliers is generalized by the
Karush–Kuhn–Tucker conditions In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be o ...
, which can also take into account inequality constraints of the form h(\mathbf) \leq c for a given constant c.


Statement

The following is known as the Lagrange multiplier theorem. Let f\colon\mathbb^n \rightarrow \mathbb be the objective function, g\colon\mathbb^n \rightarrow \mathbb^c be the constraints function, both belonging to C^1 (that is, having continuous first derivatives). Let x^* be an optimal solution to the following optimization problem such that \operatorname (Dg(x^*)) = c < n (here Dg(x^*) denotes the matrix of partial derivatives, \left \right/math>): : \text\ f(x) : \text\ g(x) = 0 Then there exists a unique Lagrange multiplier \lambda^* \in \mathbb^c such that Df(x^*) = \lambda^Dg(x^*). The Lagrange multiplier theorem states that at any local maximum (or minimum) of the function evaluated under the equality constraints, if constraint qualification applies (explained below), then the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
of the function (at that point) can be expressed as a linear combination of the gradients of the constraints (at that point), with the Lagrange multipliers acting as
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
s. This is equivalent to saying that any direction perpendicular to all gradients of the constraints is also perpendicular to the gradient of the function. Or still, saying that the
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity s ...
of the function is 0 in every feasible direction.


Single constraint

For the case of only one constraint and only two choice variables (as exemplified in Figure 1), consider the
optimization problem In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables ...
: \text\ f(x,y) : \text\ g(x,y) = 0 (Sometimes an additive constant is shown separately rather than being included in g, in which case the constraint is written g(x,y)=c , as in Figure 1.) We assume that both f and g have continuous first
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
s. We introduce a new variable ( \lambda ) called a Lagrange multiplier (or Lagrange undetermined multiplier) and study the Lagrange function (or Lagrangian or Lagrangian expression) defined by : \mathcal(x,y,\lambda) = f(x,y) - \lambda g(x,y), where the \lambda term may be either added or subtracted. If f(x_0, y_0) is a maximum of f(x,y) for the original constrained problem and \nabla g(x_0,y_0) \ne 0, then there exists \lambda_0 such that ( x_0, y_0, \lambda_0 ) is a
stationary point In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" inc ...
for the Lagrange function (stationary points are those points where the first partial derivatives of \mathcal are zero). The assumption \nabla g \ne 0 is called constraint qualification. However, not all stationary points yield a solution of the original problem, as the method of Lagrange multipliers yields only a
necessary condition In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
for optimality in constrained problems. Sufficient conditions for a minimum or maximum also exist, but if a particular
candidate solution In mathematical optimization, a feasible region, feasible set, search space, or solution space is the set of all possible points (sets of values of the choice variables) of an optimization problem that satisfy the problem's constraints, potent ...
satisfies the sufficient conditions, it is only guaranteed that that solution is the best one ''locally'' – that is, it is better than any permissible nearby points. The ''global'' optimum can be found by comparing the values of the original objective function at the points satisfying the necessary and locally sufficient conditions. The method of Lagrange multipliers relies on the intuition that at a maximum, cannot be increasing in the direction of any such neighboring point that also has . If it were, we could walk along to get higher, meaning that the starting point wasn't actually the maximum. Viewed in this way, it is an exact analogue to testing if the derivative of an unconstrained function is 0, that is, we are verifying that the directional derivative is 0 in any relevant (viable) direction. We can visualize
contour Contour may refer to: * Contour (linguistics), a phonetic sound * Pitch contour * Contour (camera system), a 3D digital camera system * Contour, the KDE Plasma 4 interface for tablet devices * Contour line, a curve along which the function ha ...
s of given by for various values of , and the contour of given by . Suppose we walk along the contour line with . We are interested in finding points where almost does not change as we walk, since these points might be maxima. There are two ways this could happen: # We could touch a contour line of , since by definition does not change as we walk along its contour lines. This would mean that the tangents to the contour lines of and are parallel here. # We have reached a "level" part of , meaning that does not change in any direction. To check the first possibility (we touch a contour line of ), notice that since the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
of a function is perpendicular to the contour lines, the tangents to the contour lines of and are parallel if and only if the gradients of and are parallel. Thus we want points where and :\nabla_ f = \lambda \, \nabla_ g, for some \lambda where : \nabla_ f= \left( \frac, \frac \right), \qquad \nabla_ g= \left( \frac, \frac \right) are the respective gradients. The constant \lambda is required because although the two gradient vectors are parallel, the magnitudes of the gradient vectors are generally not equal. This constant is called the Lagrange multiplier. (In some conventions \lambda is preceded by a minus sign). Notice that this method also solves the second possibility, that is level: if is level, then its gradient is zero, and setting \lambda = 0 is a solution regardless of \nabla_g. To incorporate these conditions into one equation, we introduce an auxiliary function : \mathcal(x,y,\lambda) = f(x,y) - \lambda g(x,y), and solve : \nabla_ \mathcal(x, y, \lambda) = 0. Note that this amounts to solving three equations in three unknowns. This is the method of Lagrange multipliers. Note that \nabla_ \mathcal(x, y, \lambda) = 0 implies g(x,y) = 0 , as the partial derivative of \mathcal with respect to \lambda is -g(x,y), which clearly is zero if and only if g(x,y) = 0. To summarize : \nabla_ \mathcal(x, y, \lambda) = 0 \iff \begin \nabla_ f(x , y) = \lambda \, \nabla_ g(x , y)\\ g(x,y) = 0 \end The method generalizes readily to functions on n variables : \nabla_ \mathcal(x_1, \dots, x_n, \lambda)=0 which amounts to solving equations in unknowns. The constrained extrema of are '' critical points'' of the Lagrangian \mathcal, but they are not necessarily ''local extrema'' of \mathcal (see
Example 2 Example may refer to: * '' exempli gratia'' (e.g.), usually read out in English as "for example" * .example, reserved as a domain name that may not be installed as a top-level domain of the Internet ** example.com, example.net, example.org, ex ...
below). One may reformulate the Lagrangian as a
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
, in which case the solutions are local minima for the Hamiltonian. This is done in
optimal control Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and ...
theory, in the form of
Pontryagin's minimum principle Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It states that it ...
. The fact that solutions of the Lagrangian are not necessarily extrema also poses difficulties for numerical optimization. This can be addressed by computing the ''magnitude'' of the gradient, as the zeros of the magnitude are necessarily local minima, as illustrated in the numerical optimization example.


Multiple constraints

The method of Lagrange multipliers can be extended to solve problems with multiple constraints using a similar argument. Consider a
paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plane ...
subject to two line constraints that intersect at a single point. As the only feasible solution, this point is obviously a constrained extremum. However, the
level set In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is calle ...
of f is clearly not parallel to either constraint at the intersection point (see Figure 3); instead, it is a linear combination of the two constraints' gradients. In the case of multiple constraints, that will be what we seek in general: the method of Lagrange seeks points not at which the gradient of f is multiple of any single constraint's gradient necessarily, but in which it is a linear combination of all the constraints' gradients. Concretely, suppose we have M constraints and are walking along the set of points satisfying g_i(\mathbf)=0, i=1, \dots, M. Every point \mathbf on the contour of a given constraint function g_i has a space of allowable directions: the space of vectors perpendicular to \nabla g_i(\mathbf). The set of directions that are allowed by all constraints is thus the space of directions perpendicular to all of the constraints' gradients. Denote this space of allowable moves by A and denote the span of the constraints' gradients by S. Then A = S^, the space of vectors perpendicular to every element of S. We are still interested in finding points where f does not change as we walk, since these points might be (constrained) extrema. We therefore seek \mathbf such that any allowable direction of movement away from \mathbf is perpendicular to \nabla f(\mathbf) (otherwise we could increase f by moving along that allowable direction). In other words, \nabla f(\mathbf) \in A^ = S. Thus there are scalars \lambda_1, \lambda_2,....\lambda_M such that : \nabla f(\mathbf) = \sum_^M \lambda_k \, \nabla g_k (\mathbf) \quad \iff \quad \nabla f(\mathbf) - \sum_^M = 0. These scalars are the Lagrange multipliers. We now have M of them, one for every constraint. As before, we introduce an auxiliary function : \mathcal\left( x_1,\ldots , x_n, \lambda_1, \ldots, \lambda _M \right) = f\left( x_1, \ldots, x_n \right) - \sum\limits_^M and solve : \nabla_ \mathcal(x_1, \ldots , x_n, \lambda_1, \ldots, \lambda _M)=0 \iff \begin \nabla f(\mathbf) - \sum_^M = 0\\ g_1(\mathbf) = \cdots = g_M(\mathbf) = 0 \end which amounts to solving n+M equations in n+M unknowns. The constraint qualification assumption when there are multiple constraints is that the constraint gradients at the relevant point are linearly independent.


Modern formulation via differentiable manifolds

The problem of finding the local maxima and minima subject to constraints can be generalized to finding local maxima and minima on a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
M. In what follows, it is not necessary that M be a Euclidean space, or even a Riemannian manifold. All appearances of the gradient \nabla (which depends on a choice of Riemannian metric) can be replaced with the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
d.


Single constraint

Let M be a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
of dimension m. Suppose that we wish to find the stationary points x of a smooth function f:M\to\R when restricted to the submanifold N defined by g(x)=0, where g:M\to\R is a smooth function for which 0 is a
regular value In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion ...
. Let df and dg be the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
s. Stationarity for the restriction f, _ at x\in N means d(f, _N)_x=0. Equivalently, the kernel \ker(df_x) contains T_x N=\ker(dg_x). In other words, df_x and dg_x are proportional 1-forms. For this it is necessary and sufficient that the following system of m(m-1)/2 equations holds: :df_x \wedge dg_x = 0 \in \Lambda^2(T^_x M) where \wedge denotes the
exterior product In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of th ...
. The stationary points x are the solutions of the above system of equations plus the constraint g(x)=0. Note that the \tfrac m(m-1) equations are not independent, since the left-hand side of the equation belongs to the subvariety of \Lambda^(T^*_x M) consisting of decomposable elements. In this formulation, it is not necessary to explicitly find the Lagrange multiplier, a number \lambda such that df_x = \lambda \, dg_x.


Multiple constraints

Let M and f be as in the above section regarding the case of a single constraint. Rather than the function g described there, now consider a smooth function G:M\to \R^p (p>1), with component functions g_i: M \to\R, for which 0\in\R^p is a
regular value In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion ...
. Let N be the submanifold of M defined by G(x)=0. x is a stationary point of f, _ if and only if \ker(df_x) contains \ker(dG_x). For convenience let L_x=df_x and K_x=dG_x, where dG denotes the tangent map or Jacobian TM\to T\R^p. The subspace \ker(K_x) has dimension smaller than that of \ker(L_x), namely \dim(\ker(L_x))=n-1 and \dim(\ker(K_x))=n-p. \ker(K_x) belongs to \ker(L_x) if and only if L_x \in T^*_x M belongs to the image of K^*_x:\R^\to T^*_x M. Computationally speaking, the condition is that L_x belongs to the row space of the matrix of K_x, or equivalently the column space of the matrix of K^*_x (the transpose). If \omega_x \in \Lambda^(T^*_x M) denotes the exterior product of the columns of the matrix of K^*_x, the stationary condition for f, _ at x becomes :L_x \wedge \omega_x = 0 \in \Lambda^ \left (T^*_x M \right ) Once again, in this formulation it is not necessary to explicitly find the Lagrange multipliers, the numbers \lambda_1,\ldots,\lambda_p such that :df_x = \sum_^p \lambda_i d(g_i)_x.


Interpretation of the Lagrange multipliers

In this section, we modify the constraint equations from the form g_i() = 0 to the form g_i() = c_i, where the c_i are ''m'' real constants that are considered to be additional arguments of the Lagrangian expression \mathcal. Often the Lagrange multipliers have an interpretation as some quantity of interest. For example, by parametrising the constraint's contour line, that is, if the Lagrangian expression is : \begin & \mathcal(x_1, x_2, \ldots;\lambda_1, \lambda_2, \ldots; c_1, c_2, \ldots) \\ pt= & f(x_1, x_2, \ldots) + \lambda_1(c_1-g_1(x_1, x_2, \ldots))+\lambda_2(c_2-g_2(x_1, x_2, \dots))+\cdots \end then :\frac = \lambda_k. So, is the rate of change of the quantity being optimized as a function of the constraint parameter. As examples, in
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lou ...
the equations of motion are derived by finding stationary points of the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
, the time integral of the difference between kinetic and potential energy. Thus, the force on a particle due to a scalar potential, , can be interpreted as a Lagrange multiplier determining the change in action (transfer of potential to kinetic energy) following a variation in the particle's constrained trajectory. In control theory this is formulated instead as
costate equations The costate equation is related to the state equation used in optimal control. It is also referred to as auxiliary, adjoint, influence, or multiplier equation. It is stated as a vector of first order differential equations : \dot^(t)=-\frac where ...
. Moreover, by the
envelope theorem In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. As we change parameters of the objective, the envelope theorem shows that, ...
the optimal value of a Lagrange multiplier has an interpretation as the marginal effect of the corresponding constraint constant upon the optimal attainable value of the original objective function: if we denote values at the optimum with an asterisk, then it can be shown that :\frac = \lambda_k^*. For example, in economics the optimal profit to a player is calculated subject to a constrained space of actions, where a Lagrange multiplier is the change in the optimal value of the objective function (profit) due to the relaxation of a given constraint (e.g. through a change in income); in such a context is the
marginal cost In economics, the marginal cost is the change in the total cost that arises when the quantity produced is incremented, the cost of producing additional quantity. In some contexts, it refers to an increment of one unit of output, and in others it r ...
of the constraint, and is referred to as the
shadow price A shadow price is the monetary value assigned to an abstract or intangible commodity which is not traded in the marketplace. This often takes the form of an externality. Shadow prices are also known as the recalculation of known market prices in o ...
.


Sufficient conditions

Sufficient conditions for a constrained local maximum or minimum can be stated in terms of a sequence of principal minors (determinants of upper-left-justified sub-matrices) of the bordered
Hessian matrix In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
of second derivatives of the Lagrangian expression.


Examples


Example 1


Example 1a

Suppose we wish to maximize f(x,y)=x+y subject to the constraint x^2+y^2=1. The
feasible set In mathematical optimization, a feasible region, feasible set, search space, or solution space is the set of all possible points (sets of values of the choice variables) of an optimization problem that satisfy the problem's constraints, potent ...
is the unit circle, and the
level set In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is calle ...
s of are diagonal lines (with slope −1), so we can see graphically that the maximum occurs at \left(\tfrac,\tfrac\right), and that the minimum occurs at \left(-\tfrac,-\tfrac\right). For the method of Lagrange multipliers, the constraint is :g(x,y)=x^2+y^2-1=0, hence the Lagrangian function, :\begin \mathcal(x, y, \lambda) &= f(x,y) + \lambda \cdot g(x,y) \\ pt&= x+y + \lambda (x^2 + y^2 - 1), \end is a function that is equivalent to f(x,y) when g(x,y) is set to 0. Now we can calculate the gradient: :\begin \nabla_ \mathcal(x , y, \lambda) &= \left( \frac, \frac, \frac \right ) \\ pt&= \left ( 1 + 2 \lambda x, 1 + 2 \lambda y, x^2 + y^2 -1 \right) \,\color \end and therefore: :\nabla_ \mathcal(x , y, \lambda)=0 \quad \Leftrightarrow \quad \begin 1 + 2 \lambda x = 0 \\ 1 + 2 \lambda y = 0 \\ x^2 + y^2 -1 = 0 \end Notice that the last equation is the original constraint. The first two equations yield :x= y = - \frac, \qquad \lambda \neq 0. By substituting into the last equation we have: :\frac+\frac - 1=0, so :\lambda = \pm \frac, which implies that the stationary points of \mathcal are :\left(\tfrac,\tfrac, -\tfrac\right), \qquad \left(-\tfrac, -\tfrac, \tfrac\right). Evaluating the objective function at these points yields :f\left(\tfrac,\tfrac\right)=\sqrt, \qquad f\left(-\tfrac, -\tfrac\right)=-\sqrt. Thus the constrained maximum is \sqrt and the constrained minimum is -\sqrt.


Example 1b

Now we modify the objective function of Example 1a so that we minimize f(x,y)=(x+y)^2 instead of f(x,y)=x+y, again along the circle g(x,y)=x^2+y^2-1=0. Now the level sets of f are still lines of slope −1, and the points on the circle tangent to these level sets are again (\sqrt/2,\sqrt/2) and (-\sqrt/2,-\sqrt/2). These tangency points are maxima of f. On the other hand, the minima occur on the level set for f = 0 (since by its construction f cannot take negative values), at (\sqrt/2,-\sqrt/2) and (-\sqrt/2, \sqrt/2), where the level curves of f are not tangent to the constraint. The condition that \nabla_\left(f(x,y)+\lambda\cdot g(x,y)\right)=0 correctly identifies all four points as extrema; the minima are characterized in by \lambda =0 and the maxima by \lambda =-2.


Example 2

This example deals with more strenuous calculations, but it is still a single constraint problem. Suppose one wants to find the maximum values of : f(x, y) = x^2 y with the condition that the x- and y-coordinates lie on the circle around the origin with radius \sqrt3. That is, subject to the constraint : g(x,y) = x^2 + y^2 - 3 = 0. As there is just a single constraint, there is a single multiplier, say \lambda. The constraint g(x,y) is identically zero on the circle of radius \sqrt3. Any multiple of g(x,y) may be added to g(x,y) leaving g(x,y) unchanged in the region of interest (on the circle where our original constraint is satisfied). Applying the ordinary Lagrange multiplier method yields :\begin \mathcal(x, y, \lambda) &= f(x,y) + \lambda \cdot g(x, y) \\ &= x^2y + \lambda (x^2 + y^2 - 3), \end from which the gradient can be calculated: :\begin \nabla_ \mathcal(x , y, \lambda) &= \left ( \frac, \frac, \frac \right ) \\ &= \left ( 2 x y + 2 \lambda x, x^2 + 2 \lambda y, x^2 + y^2 -3 \right ). \end And therefore: :\nabla_ \mathcal(x , y, \lambda)=0 \quad \iff \quad \begin 2 x y + 2 \lambda x = 0 \\ x^2 + 2 \lambda y = 0 \\ x^2 + y^2 - 3 = 0 \end \quad \iff \quad \begin x (y + \lambda) = 0 & \text \\ x^2 = -2 \lambda y & \text \\ x^2 + y^2 = 3 & \text \end (iii) is just the original constraint. (i) implies x=0 ''or'' \lambda=-y. If x=0 then y = \pm \sqrt by (iii) and consequently \lambda=0 from (ii). If \lambda=-y, substituting this into (ii) yields x^2=2y^2. Substituting this into (iii) and solving for y gives y=\pm1. Thus there are six critical points of \mathcal: : (\sqrt,1,-1); \quad (-\sqrt,1,-1); \quad (\sqrt,-1,1); \quad (-\sqrt,-1,1); \quad (0,\sqrt, 0); \quad (0,-\sqrt, 0). Evaluating the objective at these points, one finds that : f(\pm\sqrt,1) = 2; \quad f(\pm\sqrt,-1) = -2; \quad f(0,\pm \sqrt)=0. Therefore, the objective function attains the
global maximum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...
(subject to the constraints) at (\pm\sqrt,1) and the
global minimum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
at (\pm\sqrt,-1). The point (0,\sqrt) is a
local minimum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
of f and (0,-\sqrt) is a
local maximum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
of f, as may be determined by consideration of the
Hessian matrix In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
of \mathcal(x,y,0). Note that while (\sqrt, 1, -1) is a critical point of \mathcal, it is not a local extremum of \mathcal. We have :\mathcal \left (\sqrt + \varepsilon, 1, -1 + \delta \right ) = 2 + \delta \left( \varepsilon^2 + \left (2\sqrt \right)\varepsilon \right). Given any neighbourhood of (\sqrt, 1, -1), one can choose a small positive \varepsilon and a small \delta of either sign to get \mathcal values both greater and less than 2. This can also be seen from the Hessian matrix of \mathcal evaluated at this point (or indeed at any of the critical points) which is an
indefinite matrix In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a co ...
. Each of the critical points of \mathcal is a
saddle point In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function ...
of \mathcal.


Example 3:

Entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...

Suppose we wish to find the
discrete probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
on the points \ with maximal
information entropy In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable X, which takes values in the alphabet \ ...
. This is the same as saying that we wish to find the least structured probability distribution on the points \. In other words, we wish to maximize the
Shannon entropy Shannon may refer to: People * Shannon (given name) * Shannon (surname) * Shannon (American singer), stage name of singer Shannon Brenda Greene (born 1958) * Shannon (South Korean singer), British-South Korean singer and actress Shannon Arrum Wi ...
equation: :f(p_1,p_2,\ldots,p_n) = -\sum_^n p_j\log_2 p_j. For this to be a probability distribution the sum of the probabilities p_i at each point x_i must equal 1, so our constraint is: :g(p_1,p_2,\ldots,p_n)=\sum_^n p_j = 1. We use Lagrange multipliers to find the point of maximum entropy, \vec^, across all discrete probability distributions \vec on \. We require that: :\left.\frac(f+\lambda (g-1))\_=0, which gives a system of equations, k = 1,\ldots,n, such that: :\left.\frac\left\\_ = 0. Carrying out the differentiation of these equations, we get :-\left(\frac+\log_2 p^*_k \right) + \lambda = 0. This shows that all p^*_k are equal (because they depend on only). By using the constraint :\sum_j p_j =1, we find :p^*_k = \frac. Hence, the uniform distribution is the distribution with the greatest entropy, among distributions on points.


Example 4: Numerical optimization

The critical points of Lagrangians occur at
saddle point In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function ...
s, rather than at local maxima (or minima). Unfortunately, many numerical optimization techniques, such as
hill climbing numerical analysis, hill climbing is a mathematical optimization technique which belongs to the family of local search. It is an iterative algorithm that starts with an arbitrary solution to a problem, then attempts to find a better solution ...
,
gradient descent In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the ...
, some of the
quasi-Newton method Quasi-Newton methods are methods used to either find zeroes or local maxima and minima of functions, as an alternative to Newton's method. They can be used if the Jacobian or Hessian is unavailable or is too expensive to compute at every iteration. ...
s, among others, are designed to find local maxima (or minima) and not saddle points. For this reason, one must either modify the formulation to ensure that it's a minimization problem (for example, by extremizing the square of the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
of the Lagrangian as below), or else use an optimization technique that finds
stationary points In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" in ...
(such as
Newton's method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
without an extremum seeking
line search In optimization, the line search strategy is one of two basic iterative approaches to find a local minimum \mathbf^* of an objective function f:\mathbb R^n\to\mathbb R. The other approach is trust region. The line search approach first finds a d ...
) and not necessarily extrema. As a simple example, consider the problem of finding the value of that minimizes f(x)=x^2, constrained such that x^2=1. (This problem is somewhat untypical because there are only two values that satisfy this constraint, but it is useful for illustration purposes because the corresponding unconstrained function can be visualized in three dimensions.) Using Lagrange multipliers, this problem can be converted into an unconstrained optimization problem: :\mathcal(x,\lambda)=x^2+\lambda(x^2-1). The two critical points occur at saddle points where and . In order to solve this problem with a numerical optimization technique, we must first transform this problem such that the critical points occur at local minima. This is done by computing the magnitude of the gradient of the unconstrained optimization problem. First, we compute the partial derivative of the unconstrained problem with respect to each variable: : \begin & \frac=2x+2x\lambda \\ pt& \frac=x^2-1. \end If the target function is not easily differentiable, the differential with respect to each variable can be approximated as : \begin \frac\approx\frac, \\ pt\frac\approx\frac, \end where \varepsilon is a small value. Next, we compute the magnitude of the gradient, which is the square root of the sum of the squares of the partial derivatives: : \begin h(x,\lambda) & = \sqrt \\ pt& \approx\sqrt. \end (Since magnitude is always non-negative, optimizing over the squared-magnitude is equivalent to optimizing over the magnitude. Thus, the ''square root" may be omitted from these equations with no expected difference in the results of optimization.) The critical points of occur at and , just as in \mathcal. Unlike the critical points in \mathcal, however, the critical points in occur at local minima, so numerical optimization techniques can be used to find them.


Applications


Control theory

In
optimal control Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and ...
theory, the Lagrange multipliers are interpreted as costate variables, and Lagrange multipliers are reformulated as the minimization of the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
, in
Pontryagin's minimum principle Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It states that it ...
.


Nonlinear programming

The Lagrange multiplier method has several generalizations. In
nonlinear programming In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or sta ...
there are several multiplier rules, e.g. the Carathéodory–John Multiplier Rule and the Convex Multiplier Rule, for inequality constraints.


Power systems

Methods based on Lagrange multipliers have applications in
power systems An electric power system is a network of electrical components deployed to supply, transfer, and use electric power. An example of a power system is the electrical grid that provides power to homes and industries within an extended area. The e ...
, e.g. in distributed-energy-resources (DER) placement and load shedding.


See also

*
Adjustment of observations Least-squares adjustment is a model for the solution of an overdetermined system of equations based on the principle of least squares of observation residuals. It is used extensively in the disciplines of surveying, geodesy, and photogrammetry—th ...
* Duality *
Gittins index The Gittins index is a measure of the reward that can be achieved through a given stochastic process with certain properties, namely: the process has an ultimate termination state and evolves with an option, at each intermediate state, of terminati ...
*
Karush–Kuhn–Tucker conditions In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be o ...
: generalization of the method of Lagrange multipliers *
Lagrange multipliers on Banach spaces In the field of calculus of variations in mathematics, the method of Lagrange multipliers on Banach spaces can be used to solve certain infinite-dimensional constrained optimization problems. The method is a generalization of the classical metho ...
: another generalization of the method of Lagrange multipliers *
Lagrange multiplier test In statistics, the score test assesses constraints on statistical parameters based on the gradient of the likelihood function—known as the ''score''—evaluated at the hypothesized parameter value under the null hypothesis. Intuitively, if the ...
in maximum likelihood estimation *
Lagrangian relaxation In the field of mathematical optimization, Lagrangian relaxation is a relaxation method which approximates a difficult problem of constrained optimization by a simpler problem. A solution to the relaxed problem is an approximate solution to the o ...


References


Further reading

* * * * * * *


External links

Exposition
kipid's blog - Method of Lagrange multipliers
(plus a brief discussion of Lagrange multipliers in the
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
as used in physics)
Lagrange Multipliers for Quadratic Forms With Linear Constraints
by Kenneth H. Carpenter For additional text and interactive applets

* ttp://nlp.cs.berkeley.edu/tutorials/lagrange-multipliers.pdf Lagrange Multipliers without Permanent ScarringExplanation with focus on the intuition by Dan Klein
Geometric Representation of Method of Lagrange Multipliers
Provides compelling insight in 2 dimensions that at a minimizing point, the direction of steepest descent must be perpendicular to the tangent of the constraint curve at that point. eeds InternetExplorer/Firefox/Safari''Mathematica'' demonstration by Shashi Sathyanarayana
AppletMIT OpenCourseware Video Lecture on Lagrange Multipliers from Multivariable Calculus courseSlides accompanying Bertsekas's nonlinear optimization text
with details on Lagrange multipliers (lectures 11 and 12)
Geometric idea behind Lagrange multipliersMATLAB example of using Lagrange Multipliers in Optimization
{{authority control Multivariable calculus Mathematical optimization Mathematical and quantitative methods (economics)